Recently, Geometric Algebra (GA) has attracted more and more attention in the field of signal and image processing. GA can treat multi-dimensional signals in a holistic way to keep the correlations among multiple dimensions and avoid information loss. So when traditional signal and image processing algorithms are redefined in GA space, they will be more powerful and achieve better performance in multi-dimensional signal processing. In this paper, we provide a comprehensive survey covering various GA-based algorithms. In particular, we first review the mathematic theories of GA and Reduced Geometric Algebra (RGA). Then, advanced GA-based algorithms are elaborately analyzed and compared, including GA-based Sparse representation model, GA-based Dictionary Learning method, Clifford Support Vector Machine, GA-based Feature extraction algorithms, GA-based adaptive filtering algorithms, GA-based Fourier-type transform, and GA-based edge detection algorithm. Finally, we discuss several open issues and challenges of GA, and point out possible research directions in the future.INDEX TERMS Geometric algebra (GA), multivectors, multi-dimensional signals, image processing.

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